Large Deviation For Outlying Coordinates in Beta Ensembles

arXiv:1304.1446v3 [math.PR] 12 Jan 2014

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES THOMAS BLOOM*

May 28, 2013

Abstract. For Y a subset of the complex plane, a β ensemble is a sequence of probability measures P robn,β,Q on Y n for n = 1, 2, . . . depending on a positive real parameter β and a real-valued continuous function Q on Y. We consider the associated sequence of probability measures on Y where the probability of a subset W of Y is given by the probability that at least one coordinate of Y n belongs to W. With appropriate restrictions on Y, Q we prove a large deviation principle for this sequence of probability measures. This extends a result of Borot-Guionnet to subsets of the complex plane and to β ensembles defined with measures using a BernsteinMarkov condition.

1. introduction β ensembles are generalizations of the joint probability distributions of the eigenvalues of the classical matrix ensembles. They are defined as follows: let Y be a closed subset of C, Q a real-valued continuous function on Y and β > 0. Consider the family of probability distributions P robn,β,Q for n = 1, 2, ... defined on Y n by: (1.1)

P robn,β,Q =

An,β,Q(z) dτ (z) Zn,β,Q

where (1.2)

An,β,Q (z) := |D(z1 , . . . , zn )|β exp(−2n[Q(z1 ) + . . . + Q(zn )]), Y D(z1 , . . . , zn ) = (zi − zj ) 1≤i 0. In this paper we are concerned with the behaviour of a single eigenvalue/particle/coordinate. We will establish a large deviation principle for the coordinate z1 ∈ Y under the probability distributions (1.1). That is we will prove a large deviation principle for the countable family of probability distributions on Y given by, for W an open subset of Y Z Z 1 (1.4) ψn (W ) = P robn,β,Q{z1 ∈ W } = An,β,Q (z)dτ (z) Zn,β,Q W Y n−1 for n = 1, 2, . . . . The main results of this paper are Theorems 6.2 and 7.3 which show that the sequence of probability measures ψn satisfies a large deviation principle with speed n and rate function Z (1.5) JY,β,Q(z1 ) = 2Q(z1 ) − β( log |z1 − t|dµY,β,Q + ρ) Y

where µY,β,Q is the weighted equilibrium measure (see (2.3)) and ρ is a constant (see(2.5)). This l.d.p. implies that the probability of finding an eigenvalue/ particle/coordinate outside a neighbourhood of the support of the weighted equilibrium measure is asymptotic to e−nc for some c > 0. Theorem 6.2 handles the case when Y is compact and Theorem 7.3 the unbounded case. Theorem 6.2 is valid under the following hypotheses. (We let SR denote the support of the weighted equilibrium measure and SR∗ the points where Jn,β,Q(z1 ) = 0) (1) Y is a regular set (in the sense of potential theory.) (2) (Hypothesis 3.9) τ satisfies the weighted BM inequality on any compact neighbourhood of SR∗ . (3) (Hypothesis 6.1) SR∗ = SR . Now (1) ensures that the weighted Green function (see (2.2)) is continuous. (2) is always satisfied by Lebesgue measure on R or C (see section 3). (3) is the assumption that the rate function is strictly positive outside SR . In [9] the corresponding assumption is referred to as control of large deviation. The unbounded case, Theorem 7.3, requires additional assumptions. Theorem 7.3 is valid under the following assumptions: Condition (1)

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THOMAS BLOOM*

above, is replaced by the hypothesis that the intersection of Y with sufficiently large discs centre the origin is regular. (2) and (3) above must hold. The additional conditions are on the growth of τ and R = 2Q/β. Namely: (4) (see(2.1)) For some b > 0, lim|z|→∞R(R(z) − (1 + b) log |z|) = +∞. (5) (Hypothesis 7.1) For some a > 0, Y dτ /|z|a < +∞. A key step is theorem 5.1 where we prove a result on the asymptotics of normalizing constants, proved in [9] for subsets of R and used previously as an assumption in [1]. Our methods use polynomial estimates and potential theory but not the large deviation principle for the normalized counting measures of a random point. We will specifically use weighted potential theory (see [19]), since the probability distributions given by (1.1) are, in each variable, a power of the absolute value of a weighted polynomial (see section 2). These results extend a result of [9] to appropriate subsets of the plane and, in addition, the measure τ used to define the ensemble can be more general than Lebesgue measure: it need only satisfy the Bernstein-Markov condition given by Hypothesis 3.9. The rate function is independent of τ as long as τ satisfies Hypothesis 3.9. We let C, c denote positive constants which may vary from line to line. All measures are positive Borel measures. 2. polynomial estimates We will list some basic results of weighted potential theory (see [19]). Let Y be a closed set in the plane and R be a continuous (real-valued) function on Y . If Y is unbounded , R is assumed to be superlogarithmic i.e., for some b > 0 (2.1)

lim (R(z) − (1 + b) log |z|) = +∞.

|z|→∞

For r > 0 we let Yr := {z ∈ Y ||z| ≤ r}. We will assume that Yr is a regular set (in the sense of potential theory) for all r sufficiently large. We recall that a compact set is regular if it is regular for the exterior Dirichlet problem [17] or equivalently if the unweighted Green function (i.e. R ≡ 0 in (2.2) below) is continuous on C. The weighted Green function of Y with respect to R ([19], Appendix B) is denoted by VY,R . It is defined by (2.2)

VY,R (z) = sup{u(z)|u is subharmonic on C, u ≤ R on Y and,

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES

5

u ≤ log+ |z| + C, where C is a constant depending on u}. VYr ,R is continuous for r sufficiently large ([18], prop 2.16) since Yr was assumed to be regular for r sufficiently large and any regular compact set in the plane is locally regular ([17]). Thus VY,R is continuous since VY,R = VYr ,R for r sufficiently large ([19], Appendix B, lemma 2.2). The logarithmic capacity c(E) of a Borel subset E ⊂ C is defined by log c(E) = sup{ ν

Z Z

log |z − t|dν(z)dν(t)}

over all Borel probability measures ν with compact support in E. Regular sets are of positive capacity and so Y is of positive capacity. The function e−R is thus strictly positive on a set of positive capacity and so is an admissible weight in the terminology of [19]. Thus Chapter I, Theorem 1.3 of [19] holds. The weighted equilibrium measure is denoted µY,R. It has compact support and is the unique minimizer of Z Z Z (2.3) E(ν) = − log |z − t|dν(z)dν(t) + 2 R(z)dν(z) over all measures ν ∈ M(Y ) where M(Y ) denotes the probability measures on Y ([19], Chapter I, Theorem 1.3). Now, by ([19], Appendix B, Lemma 2.4) Z (2.4) VY,R (z) = log |z − t|dµY,R(t) + ρ Y

where ρ is the Robin constant given by ρ = lim (VY,R (z) − log |z|).

(2.5)

|z|→∞

It is also known that E(µY,R ) > −∞ and ([19], Chapter I, Theorem 3(d)) Z ρ = E(µY,R ) − R(z)µY,R . Y

SR∗

We let := {z ∈ Y |VY,R = R} and SR := supp(µY,R). SR and SR∗ are compact and in general SR ⊂ SR∗ ([19], Chapter I, Theorem 1.3(e)). SR and SR∗ depend on Y although our notation does not explicitly indicate this. However, for r sufficiently large SR and SR∗ are independent of Yr .

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THOMAS BLOOM*

From (2.4) we have ([19], Chapter I, Theorem 1.3(f)) Z (2.6) R(z) = log |z − t|dµY,R(t) + ρ Y

for z ∈ SR . We let Pn denote the space of polynomials in the single variable z of degree ≤ n and for p ∈ Pn we refer to e−nR(z) p(z) as a weighted polynomial of degree n (the weight is the positive continuous function e−R(z) ). Now for β > 0 we set (2.7)

R(z) =

2 Q(z). β

Note that An,β,Q(z) is, in each variable, of the form |e−nR(z) p(z)|β = |e−2nQ p(z)|β and is thus the absolute value of a weighted polynomial to the β power. For this reason weighted potential theory can be used in the study of β ensembles. Equations (2.8) and (2.9) below are known estimates for weighted polynomials. By ([19], Chapter III, Corollary 2.6) the sup norm of a weighted polynomial is assumed on SR . That is (2.8)

||e−nR p(z)||Y = ||e−nR(z) p(z)||SR

for all p ∈ Pn . By ([19], Chapter I, Theorem 3.6) we have, for p a monic polynomial of degree n (2.9)

||e−nR(z) p(z)||SR ≥ e−nρ .

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES

7

For G a subset of M(Y ) we will use the notation n X ˜ n = {z = (z1 , . . . , zn ) ∈ Y n 1 δ(zj ) ∈ G}. G n j=1

Here δ denotes the Dirac delta measure.

We now restrict to the case that Y is compact and we will use the notation K in place of Y. Lemma 2.1. Given ǫ > 0, there is a neighbourhood G of µK,R in M(K) ˜ n we have (with the weak* topology) such that for (z1 , . . . , zn ) ∈ G ||e−nR(t) (t − z1 ) . . . (t − zn )||K ≤ e−n(ρ−ǫ) . Proof. The proof will be by contradiction. If not, for some ǫ > 0, no such G exists. Thus there exists a sequence of ns -tuples (z1s , . . . , zns s ) for s = 1, 2, . . . with (2.10)

lim s

ns 1 X δ(zjs ) = µK,R ns j=1

weak*, but, using (2.8), (2.11)

||e−ns R(t) (t − z1s ) . . . (t − zns s )||SR ≥ e−ns ρ ens ǫ .

Taking logarithms this may be rewritten as: (2.12)

ns

1 X

s log |t − zj | ≥ −ρ + ǫ.

− R(t) + SR ns j=1

It follows from (2.10) that, using ([19], Chapter 0, Theorem 1.4), we have Z ns 1 X s (2.13) lim sup log |t − zj | ≤ log |t − ξ|dµK,R(ξ), ns j=1 s since ξ → log |t − ξ| is uppersemicontinuous. Now, (2.13) is a pointwise lim sup and to obtain a contradiction we will require a uniform lim sup for t ∈ SR .

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THOMAS BLOOM*

Since ξ → log |t − ξ| is subharmonic, by Hartogs’ lemma, ([15], Theorem 2.6.4) and (2.6) we have (2.14)

lim sup s

ns 1 X log |t − zjs | ≤ R(t) − ρ + ǫ/2, ns j=1

uniformly on SR . That is, for s sufficiently large, ns

1 X

s (2.15) log |t − zj | ≤ −ρ + ǫ/2,

− R(t) + ns j=1 SR which contradicts (2.12).

 The following is a corollary to the proof of Lemma 2.1 and will be used in the proof of Theorem 6.2. Corollary 2.2. Given ǫ > 0, there is a neighborhood G of µK,R in ˜ n−1 and M(K) (with the weak* topology) such that for (z2 , . . . , zn ) ∈ G w ∈ K then Z n 1 X log |w − zj | − log |w − t|dµK,R(t)) ≤ ǫ. (2.16) ( n − 1 j=2 3. Bernstein-Markov inequalities Although we ultimately will only need Bernstein-Markov inequalities for subsets of C we will consider these inequalities in CN as the C2 case is used in the proofs. Recall that all measures are positive Borel measures. Definition 3.1. Let τ be a measure on a compact set K ⊂ CN . We say τ satisfies the Bernstein-Markov (BM) inequality if, for all ǫ > 0, there exists C > 0 (independent of n, p) such that Z n |p|dτ, (3.1) ||p||K ≤ C(1 + ǫ) K

PnN ,

for all p ∈ where here total degree ≤ n on CN . .

PnN

denotes the holomorphic polynomials of

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES

9

Definition 3.2. Let τ be a measure on a compact set K ⊂ CN and R a real-valued continuous function on K. We say τ satisfies the weighted Bernstein-Markov (BM) inequality for the weight e−R if, for all ǫ > 0, there exists C > 0 (independent of n, p) such that Z −nR n e−nR |p|dτ, (3.2) ||e p||K ≤ C(1 + ǫ) K

for all p ∈

PnN .

We say that τ satisfies a strong BM inequality if it satisfies the weighted BM inequality for all weights e−R . It is known ([20], proof of Theorem 3.4.3 ) that if τ satisfies (3.1) or (3.2) then it also satisfies an Lβ version of that inequality (with, possibly, a different constant C). For example, from (3.2) Z 1 −nR n (3.3) ||e p||K ≤ C(1 + ǫ) ( (e−nR |p|)β dτ ) β . K

The following lemma will be used in the proof of Theorem 5.1. Lemma 3.3. Let K ⊂ C be regular, R a continuous real-valued continuous function on K and τ be a measure on K satisfying the weighted BM inequality for the weight e−R . Then for all monic polynomials of degree n Z (3.4) |e−2nQ pβ |dτ ≥ C(1 + ǫ)−nβ e−nρβ . K

Proof. Recall that R = 2Q/β and combine (3.2) with (2.8) and (2.9).  Conditions on a measure to satisfy the BM inequality for compact sets K ⊂ C are extensively studied in [20]. The measures termed regular in [20] coincide with measures satisfying the BM inequality on regular compact sets but the BM inequality is a more stringent condition in general (see [20], example 3.5.3). Remark 3.4. Suppose that τ satisfies the weighted BM inequality for e−R on SR . Then it follows from (3.2) and (2.8) that τ satisfies the weighted BM inequality for e−R on any compact set K ⊃ SR . Hence any measure τ which satisfies the weighted BM inequality for e on SR will satisfy Hypothesis 3.9 below i.e. τ satisfies the weighted −R

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THOMAS BLOOM*

BM inequality on any compact neighbourhood of SR . Now for every compact set K ⊂ C there exists a discrete measure ( i.e. a countable linear combination of Dirac δ measures ) on K which satisfies the strong BM inequality (see [5], Corollary 3.8 - the construction given there is, in fact, valid for every compact set). Applying this construction to SR gives a discrete measure which satisfies Hypothesis 3.9. We will show that Lebesgue measure on R or C also satisfies Hypothesis 3.9. The next proposition gives a convenient sufficient (but by no means necessary) condition for a measure τ to satisfy the BM inequality on a regular compact set K ⊂ CN (for the definition of regular compact set in CN , N > 1 see ([15], Chapter 5). Proposition 3.5. ( [7], Theorem 2.2) Let K ⊂ CN be compact and regular. Let τ be a measure on K such that for some T > 0 and each z0 ∈ K there exists r(z0 ) > 0 such that (3.5)

τ {B(z0 , r)} ≥ r T for r ≤ r(z0 )

where B(z0 , r) denotes the ball centre z0 , radius r. Then τ satisfies the BM inequality on K. We also have: Proposition 3.6. ( [3], Theorem 3.2) Let K ⊂ RN ⊂ CN . Then if τ satisfies the BM inequality on K it satisfies the strong BM inequality on K. Corollary 3.7. Let K be the closure of its interior in C and have C 1 boundary. Then Lebesgue planar measure satisfies the strong BM inequality on K. Proof. Consider K ⊂ C ∼ = R2 ⊂ C2 . Now, K is regular as a subset of C2 (this follows, for example from the accessibility condition of Plesniak,[15], Chapter 5) and any measure satisfying a strong BM inequality on K as a subset of C2 satisfies a strong BM inequality on K as a subset of C. Thus the result follows from Propositions 3.5 and 3.6.  Corollary 3.8. Let K ⊂ R be a finite union of disjoint closed intervals.Then Lebesgue measure on R satisfies the strong BM inequality on K.

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 11

Proof. K is regular as a subset of C ([20]) so it follows from Propositions 3.5 and 3.6 that Lebesgue measure on R satisfies the strong BM inequality on K.  We will use the following hypothesis in the next lemma: Hypothesis 3.9. The measure τ satisfies the weighted BM inequality for e−R on any compact neighbourhood of SR∗ . Remark 3.10. Every compact set K ⊂ C has a neighbourhood basis of compact sets which are the closure of their interiors and have C 1 boundary. Thus by Corollary 3.7, Lebesgue measure on C satisfies Hypothesis 3.9. Similarly, since every compact set K ⊂ R has a neighbourhood basis of compact sets which are a finite disjoint union of closed intervals, so by Corollary 3.8, Lebesgue measure on R satisfies Hypothesis 3.9. Lemma 3.11. Let K ⊂ C be regular, R a real-valued continuous function on K and τ be a measure on K satisfying Hypothesis 3.9. Let N be a compact neighbourhood of SR∗ . Then there is a constant c > 0 (independent of n, p) such that: Z Z −nR β −nc |e p| dτ ≤ (1 + O(e )) |e−nR p|β dτ K

N

for all p ∈ Pn . Proof. We first normalize the polynomial so that ||e−nR p||SR = 1. To prove the theorem it will suffice to show that Z (3.6) |e−nR(z) p(z)|β dτ ≤ Ce−nc , K\N

for some constants C, c > 0 and Z 1/n (3.7) lim inf |e−nR p|β dτ ≥ 1. n

K

We will use the estimate ([19], Appendix B, Theorem 2.6 (ii)) (3.8)

|e−nR(z) p(z)| ≤ ||e−nR(z) p(z)||SR exp(n(VK,R (z) − R(z))

for z ∈ K. Since N is a compact neighbourhood of SR∗ , for z ∈ K \ N, and some constant b > 0 (3.9)

VK,R (z) − R(z) ≤ −b < 0

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THOMAS BLOOM*

so |e−nR(z) p(z)| ≤ e−nb for z ∈ K \ N. Thus, Z (3.10) |e−nR(z) p(z)|β dτ ≤ Ce−nbβ K\N

for constants C, b > 0. Now, Z Z −nR(z) β (3.11) |e p(z)| dτ ≥ |e−nR(z) p(z)|β dτ K

N

and since τ satisfies the weighted BM condition on N for e−R the right hand side in (3.11) is, for any ǫ > 0 and some C > 0 (3.12)

≥ ||e−nRp||βSR Ce−ǫn = Ce−ǫn ,

establishing (3.7) and the result.  Theorem 7.2 establishes a version of this lemma for unbounded sets. 4. Johansson large deviation We will not use the l.d.p. for the normalized counting measure of a random point but a weaker result whose utility was shown by Johansson [14]. Consider (4.1)

An,β,Q (z) := |D(z1 , . . . , zn )|β exp(−2n[Q(z1 ) + . . . + Q(zn )])

where β > 0, D denotes the Vandermonde determinant Y D(z1 , . . . , zn ) = (zi − zj ), 1≤i 0 and let η , 0 < η < γ be given. We define 1

Q n2 Bη,n,β := {z ∈ K n |An,β,Q (z) ≤ γ − η}.

Then we have the following result, which we refer to as a Johansson large deviation result: Proposition 4.2. Q P robn,β,Q (Bη,n,β )

=

1 Zn,β,Q

Z

Q Bη,n,β

An,β,Q(z)dτ (z) ≤ (1 −

η n2 ) τ (K)n 2γ

for all n sufficiently large. Proof. See [8], Proposition 4.15 or [2], Theorem 4.1.



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THOMAS BLOOM*

Proposition 4.3. Let G be a neighbourhood of µK,β,Q in M(K).Then Z 1 2 An,β,Q (z)dτ ≤ O(e−cn ) Zn,β,Q K n \G˜ n for some c > 0. Proof. It follows (see [8], claim (6.16) or Proposition 7.3 of [2]) that for some η > 0, Q ˜n Bη,n,β ⊃ Kn \ G for all n sufficiently large, and so the result then follows from Proposition 4.2.  5. the normalizing constants Proposition 4.1 gives an asymptotic result for the normalizing constants Zn,β,Q. Note that if we have a sequence of continuous functions {Qn } converging uniformly to Q on K, then n12 log Zn,β,Q and 1 log Zn,β,Qn have the same limit, and in particular, this is true for n2 1 log Zn,β, nQ . We will, however, need a sharper result, namely: n2 n−1

Theorem 5.1. Let K be a regular compact subset of C, Q a real-valued continuous function on K and τ a measure on K satisfying the weighted BM inequality for e−R . Then  Z  n1 n,β,Q lim = e−ρβ . n→∞ Z nQ n−1,β, n−1

Proof. We will first prove that  n1  Z n,β,Q ≥ e−ρβ . (5.1) lim inf n→∞ Zn−1,β, nQ n−1 R Now Zn,β,Q = K n An,β,Q (z)dτ (z). We regard An,β,Q(z) as a function of z1 and we apply Lemma 3.3 to the integral in the z1 variable to obtain (5.2) Z

|D(z2 , . . . , zn )|β e−2n[Q(z2 )+...+Q(zn )] dτ (z2 ) . . . dτ (zn )

Zn,β,Q ≥ C(1+ǫ)−nβ e−nρβ

Kn

= C(1 + ǫ)−nβ e−nρβ Zn−1,β, nQ . n−1

Since ǫ > 0 is arbitrary, the estimate on the lower limit follows.

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 15

To complete the proof we must show  n1  Z n,β,Q (5.3) lim sup ≤ e−ρβ . Zn−1,β, nQ n→∞ n−1

˜ n−1 be a subset of K n−1 determined as follows: Given ǫ > 0 Let G choose a neighbourhood G of µK,β,Q so that Lemma 2.1 holds. Recall that n X ˜ n−1 = {(z2 , . . . , zn ) ∈ K n−1 | 1 δ(zj ) ∈ G}. G n − 1 j=2 We write the integral for Zn,β,Q as a sum of two integrals Zn,β,Q = I1 + I2 where I1 =

Z Z

˜ n−1 K n−1 \G

K

and I2 =

Z Z K

˜ n−1 G

An,β,Q (z)dτ (z)

An,β,Q (z)dτ (z).

Now, (5.4)

I1 =

Z Y n

|z1 − zj |β e−2nQ(z1 ) dτ (z1 )×

K j=2

Z

˜ K n−1 \G

|D(z2 , . . . , zn )|β e−2n[Q(z2 )+...+Q(zn )] dτ (z2 ) . . . dτ (zn ). n−1

Since K is compact, the first factor is O(C n ) for some C > 0. The integrand in the second factor differs from An−1,β,Q (z2 , . . . , zn ) by a 2 factor of O(C n ) for some C > 0 so the second factor is O(e−cn )Zn−1,β,Q by Proposition 4.3. Since Zn−1,β,Q = O(C n )Zn−1,β, nQ , n−1

we may conclude that 2

I1 = O(e−cn )Zn−1,β, nQ . n−1

Also (5.5)

I2 =

Z Y n K j=2

|z1 − zj |β e−2nQ(z1 ) dτ (z1 )×

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THOMAS BLOOM*

Z

˜ n−1 G

|D(z2 , . . . , zn )|β e−2n[Q(z2 )+...+Q(zn )] dτ (z2 ) . . . dτ (zn ).

We will need a more precise estimate on the first factor than the one ˜ n−1 we may use Lemma used in (5.4). Since we are integrating over G −nβ(ρ−ǫ) 2.1 on the first factor to see that it is ≤ Ce . The second factor n−1 ˜ is ≤ Zn−1,β, nQ since Gn−1 is a subset of K . Hence, given any ǫ > 0, n−1 we have for some c, C > 0 2

I1 + I2 ≤ Ce−nβ(ρ−ǫ) Zn−1,β, nQ + O(e−cn )Zn−1,β, nQ n−1

and (5.3) follows.

n−1

 6. large deviation

Given a separable, complete metric space X, a sequence of probability measures {σn } on X is said to satisfy a large deviation principle with speed n and rate function J(x) if J : X → [0, ∞] is lowersemicontinuous, {x ∈ X|J(x) ≤ l} is compact for l ≥ 0 and (i) For all closed sets F ⊂ X we have 1 lim sup log σn (F ) ≤ − inf J(x). x∈F n n (ii) For all open sets G ⊂ X we have 1 lim inf log σn (G) ≥ − inf J(x). n x∈G n If X is compact by ([10], Theorem 4.1.11) to establish the l.d.p. it suffices to show that, for all x ∈ X 1 (6.1) − J(x) = lim lim log σn (B(x, ǫ)) ǫ→0 n n where B(x, ǫ) is the ball centre x, radius ǫ. If X is non-compact, to establish the l.d.p. there is an additional condition required, termed exponential tightness, namely: For all r > 0 there is a compact set Xr ⊂ X with σn (Xr ) ≤ −r. Let K ⊂ C be a regular compact set and Q a real - valued continuous function on K. Let τ be a measure on K which satisfies the weighted BM inequality for e−R (recall that R = 2Q/β). Given the probability measures P robn,β,Q forming a β ensemble on K, we consider the countable family of probability measures {ψn }, n = 1, 2, ... on K given as follows: For W an open subset of K

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 17

(6.2) ψn (W ) = P robn,β,Q{z1 ∈ W } =

1 Zn,β,Q

Z Z W

An,β,Q (z)dτ (z). K n−1

To prove the l.d.p., we will assume that τ satisfies Hypothesis 3.9. We will also use an additional assumption. Hypothesis 6.1. (6.3)

SR = SR∗ .

Now, to prove the l.d.p. we will use equation (6.1) and so we will have to estimate 1 inf lim log ψn (W ). {W |z1 ∈W } n n We will be able to do so for z1 ∈ K \ SR∗ and z1 ∈ SR . Thus under Hypothesis 6.1 the l.d.p. will be complete. For β ensembles on R , τ the Lebesgue measure on R and Q real analytic, then results of [16] show that Hypothesis 6.1 holds for all but at most countably many values of β. A class of examples where Hypothesis 6.1 holds are provided by the computations in ([19], Chapter IV, section 6). We suppose R(z) is invariant under rotations of the plane, that R(t) is differentiable on (0, ∞) and satisfies tR′ (t) and R(t) are increasing on (0, ∞). Then SR is a disc of radius T0 given by the solution of T0 R′ (T0 ) = 1. If we take K a disc of radius > T0 , then VK,R (z) = log |z| + R(T0 ) − log T0 while R(z) > VK,R (z) for |z| > T0 so that SR∗ = SR . R(z) = |z|2 provides a specific example. It is simple to construct examples where Hypothesis 6.1 does not hold. For example, let K be a large disc with SR∗ ⊂ interior(K). Replacing R by R′ = VK,R then VK,R′ = VK,R so SR′ = SR but SR∗ ′ = K. Theorem 6.2. Let K be a regular, compact subset of C, Q a real-valued continuous function on K and τ a measure with supp(τ ) = K and satisfying Hypotheses 3.9 and 6.1. Let P robn,β,Q be a β ensemble on K. Then the sequence of measures ψn , given by ψn (W ) = P robn,β,Q{z1 ∈ W } for W an open subset of K satisfies a l.d.p. with speed n and rate function (6.4)

JK,β,Q(z1 ) = β(R(z1 ) − VK,R (z1 )) = 2Q(z1 ) − βVK,R(z1 ).

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THOMAS BLOOM*

Proof. Using (2.4) we have JK,β,Q(z1 ) = 2Q(z1 ) − β(

Z

log |z1 − t|dµK,β,Q(t) + ρ).

K

Note that JK,β,Q(z1 ) = 0 for z1 ∈ SR . Also since VK,R is continuous, the function Z (6.5) z1 → log |z1 − t|dµK,β,Q(t) K

is continuous and so is the rate function. We will use equation (6.1). Let W be an open subset of K. We will estimate the probability that a point w ∈ W. By definition, Z Z 1 (6.6) P robn,β,Q{w ∈ W } = An,β,Q (w, z2 , ..., zn )dτ. Zn,β,Q W K n−1 We will separately estimate lim supn and lim inf n of 1/n log(P robn,β,Q{w ∈ W }) We begin with the lim sup . We write the above integral as a sum of two integrals (where G is an open neighbourhood of µK,β,Q ⊂ M(K) which is to be specified). Z Z 1 An,β,Q (w, z2 , ...zn )dτ = H1 + H2 . Zn,β,Q W K n−1 Z Z 1 H1 = An,β,Q (w, z2, ...zn )dτ Zn,β,Q W K n−1 \G˜ n−1 and Z Z 1 H2 = An,β,Q (w, z2, ...zn )dτ. Zn,β,Q W G˜ n−1 2

Similar to the estimate for I1 , (see (5.4)) we have H1 = O(e−cn ). To estimate H2 we now proceed as in [1]. Let (6.7)

hn :=

Zn,β,Q Zn−1,β, nQ

.

n−1

Then (6.8)

H2 =

1 hn Zn−1,β, nQ

n−1

Z Z W

˜ n−1 G

An,β,Q (w, z2 , ...zn )dτ.

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 19

Now, given ǫ > 0, using Corollary 2.2, let G be a neighbourhood of ˜ n−1 then µK,β,Q ⊂ M(K) so that for w ∈ W and (z2 , . . . , zn ) ∈ G Z n   1 X log |w − zj | − log |w − t|dµK,β,Q(t) ≤ ǫ. (6.9) β n − 1 j=2 Taking exponentials n Y R |w − zj |β ≤ e(n−1)(ǫ+β log |w−t|dµK,β,Q (t)) . (6.10) j=2

˜ n−1 ⊂ K n−1 Also, since G (6.11) Z 1 |D(z2 , . . . , zn )|β e−2n[Q(z2 )+...+Q(zn )] dτ (z2 ) . . . dτ (zn ) ≤ 1. Zn−1,β, nQ G˜ n−1 n−1

Using Theorem 5.1 to estimate hn , the inequalities in (6.10) and (6.11), and recalling that τ (W ) 6= 0 since the support of τ is K, we have (6.12)

lim sup 1/n log(P robn,β,Q{w ∈ W }) n Z ≤ βρ + supw∈W (β log |w − t|dµK,β,Q − 2Q(w)). K

and using (6.5) we have lim sup 1/n log(P robn,β,Q{w ∈ W }) n Z ≤ βρ + β log |z1 − t|dµK,β,Q − 2Q(z1 ).

(6.13)

inf

{W |z1 ∈W }

K

Now we must deal with the lim inf . First we consider z1 ∈ / SR . Let W be a neighbourhood of z1 with W ∩ SR = ∅, and let N be a compact neighbourhood of SR such that N ∩ W = ∅. Now An,β,Q (z) is in each variable of the form e−2nQ |p|β for a polynomial p ∈ Pn . Hence by repeated use of Lemma 3.11 we have, (6.14)

Z

Kn

−cn

An,β,Q (z)dτ (z) ≤ (1 + O(e

))

Z

An,β,Q (z)dτ (z),

Nn

or (6.15)

Zn,β,Q(K) ≤ (1 + O(e−cn ))Zn,β,Q(N)

20

THOMAS BLOOM*

where K and N indicate the sets on which the normalizing constants are evaluated. Similarly, Zn−1,β, nQ (K) ≤ (1 + O(e−cn ))Zn−1,β, nQ (N).

(6.16)

n−1

Now

n−1

1 Zn,β,Q(K)

Z Z

1

≥

W

An,β,Q (w, z2 , ...zn )dτ K n−1

Z Z

An,β,Q (w, z2 , ...zn )dτ Zn,β,Q(K) W N n−1 and using (6.14), to obtain the lower bound it suffices to estimate: Z Z 1 An,β,Q (w, z2 , ...zn )dτ = Zn,β,Q (N) W N n−1 Z Z 1 An,β,Q(w, z2 , ...zn )dτ. hn (N)Zn−1,β, nQ (N) W N n−1 n−1

Given ǫ > 0 let F be a neighbourhood of µK,β,Q in M(N) such that for w ∈ W and (z2 , . . . , zn ) ∈ F˜n−1 we have, Z n  1 X  (6.17) −ǫ≤β log |w − zj | − log |w − t|dµK,β,Q(t) . n − 1 j=2 Such an F exists since log |w − t| is continuous for t ∈ N and w ∈ W. Taking exponentials n Y

(6.18)

|w − zj |β ≥ e(n−1)(−ǫ+β

R

log |w−t|dµK,β,Q (t))

.

j=2

Now we have (see (5.4)) Z 1 (6.19) |D(z2 , . . . , zn )|β e−2n[Q(z2 )+...+Q(zn )] dτ Zn−1,β, nQ (N) N n−1 \F˜n−1 n−1

2

≤ O(e−cn ). So (6.20) −cn2

1−O(e

)≤

1 Zn−1,β, nQ (N) n−1

Z

F˜n−1

|D(z2 , . . . , zn )|β e−2n[Q(z2 )+...+Q(zn )] dτ

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 21

Using the inequalities in (6.18) and (6.20), and the fact that τ (W ) 6= 0, we have, (6.21)

lim inf 1/n log(P robn,β,Q{w ∈ W }) n

≥ βρ + infw∈W (β

Z

log |w − t|dµK,β,Q − 2Q(w)).

K

By (6.5) it follows that (6.22)

inf

lim

{W |z1 ∈W } n

= β(ρ +

Z

1 log(P robn,β,Q{w ∈ W }) n

log |z1 − t|dµK,β,Q) − 2Q(z1 ). K

To complete the l.d.p. we must consider the case when z1 ∈ SR and to do so we must estimate P robn,β,Q{w ∈ W } when W ∩ SR 6= ∅. In fact, we will show that, in this case, (6.23)

lim n

1 log(P robn,β,Q{w ∈ W }) = 0. n

Now, nP robn,β,Q {w ∈ W } ≥ P robn,β,Q{at least one of w, z2 , . . . , zn ∈ W } = 1 − P robn,β,Q{each of w, z2 , . . . , zn ∈ K \ W }. Since W ∩ SR 6= ∅ the support of the weighted equilibrium measure for R on K \ W cannot be SR . This implies, using Proposition 4.1 and the minimizing property of the equilibrium measure, that 1 lim supn supK\W An,β,Q(z) n2 ≤ γ − η, for some η > 0. (Recall that 1

γ = lim supn supK An,β,Q (z) n2 .) Then one can use Proposition 4.2 to obtain (6.24)

P robn,β,Q {w ∈ W } ≥

and (6.23) follows.

1 2 (1 − O(e−cn )) n 

22

THOMAS BLOOM*

7. the unbounded case Let Y be a closed, unbounded subset of C. Let R be a continuous, real - valued, superlogarithmic function on Y. That is, for some b > 0, lim (R(z) − (1 + b) log |z|) = +∞.

|z|→∞

For r > 0 we let Yr =: {z ∈ Y ||z| ≤ r} and we assume Yr is regular for r sufficiently large. Also, for r sufficiently large VY,R = VYr ,R ([19],Appendix B, Lemma 2.2) and SR∗ ⊂ Yr . We will also denote the equilibrium measure µY,R by µY,β,Q. We will extend the l.d.p. (Theorem 6.2) from the compact case to the unbounded case. We will need an additional hypothesis on the growth of the measure τ. Hypothesis 7.1. τ is a locally finite measure on Y satisfying: Z (7.1) For some a > 0, we have dτ /|z|a < +∞ Y

We note that Lebesgue measure on R or C satisfies Hypothesis 7.1. The following theorem will extend Lemma 3.11 to the unbounded case. Both Lemma 3.11 and Theorem 7.2 are based on ([19], Chapter III, Theorem 6.1). They shows that the Lβ norm of a weighted polynomial ”lives” on SR∗ . Theorem 7.2. Let Y be a closed, unbounded subset of C with Yr regular for r sufficiently large. Let R be a superlogarithmic real-valued function on Y. Let β > 0 and let N ⊂ Y be a compact neighbourhood of SR∗ . Let τ be a measure on Y such that Hypotheses 3.9 and 7.1 are satisfied. Then there is a constant c > 0 (independent of n, p) such that: Z Z −nR β −nc |e p| dτ ≤ (1 + O(e )) |e−nR p|β dτ Y

N

for all p ∈ Pn . Proof. We need only consider the case N = Yr for r large and then the result will follow by using Lemma 3.11. We first normalize the polynomial so that ||e−nR p||SR = 1. To prove the theorem it will suffice to show that Z |e−nR(z) p(z)|β dτ ≤ Ce−nc , (7.2) Y \Yr

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 23

for some constants C, c > 0 and Z 1/n −nR β (7.3) lim inf |e p| dτ ≥ 1. n

Y

We will use the estimate ([19] , Appendix B, Theorem 2.6 (ii)) |e−nR(z) p(z)| ≤ ||e−nR(z) p(z)||SR exp(n(VY,R (z) − R(z))

(7.4) for z ∈ Y.

Using (2.1) we have, since VY,R ≤ log |z| + C for |z| large b VY,R (z) − R(z) ≤ − log |z| 2

(7.5) for |z| large . Hence, for r large Z (7.6)

−nR(z)

|e

β

p(z)| dτ ≤ C

Y \Yr

≤ Cr

− nbβ a 2

r

Z

Y \Yr

Z

dτ Y \Yr

|z|

nbβ 2

dτ ≤ Ce−nc . |z|a

for constants C, c > 0 and all n sufficiently large. Now, (7.7)

Z

Y

−nR(z)

|e

β

p(z)| dτ ≥

Z

|e−nR(z) p(z)|β dτ

Yr

and (7.3) follows from the proof of Lemma 3.11.  Let Y be a closed, unbounded subset of C with Yr regular for r sufficiently large. Let R be a superlogarithmic real-valued function on Y. We consider β ensembles P robn,β,Q on Y . Theorem 7.3. Let Y and R be as above. Assume that the measure τ satisfies Hypotheses 3.9 and 7.1 and supp(τ ) = Y. Also assume that Hypothesis 6.1 holds. Then the sequence of probability measures defined by, for W an open subset of Y , ψn (W ) = P robn,β,Q{z1 ∈ W } satisfy an l.d.p. with speed n and rate function JY,β,Q (z1 ) = β(R(z1 ) − VY,R (z1 )) = 2Q(z1 ) − VY,R (z1 ).

24

THOMAS BLOOM*

Proof. As noted in the proof of Theorem 6.2, Z JY,β,Q (z1 ) = 2Q(z1 ) − β( log |z1 − t|dµY,β,Q(t) + ρ). Y

To prove the l.d.p. will show that equation (6.1) holds, together with exponential tightness. We consider an open set W ⊂ Y. We may assume that W ⊂ Yr . Applying the results of section 6 to β ensembles on the compact set Yr for r sufficiently large, we have 1 (7.8) inf lim log P robn,β,Q{z1 ∈ W } = β(VY,R (z1 ) − R(z1 )). {W |z1 ∈W } n n We will show the same result holds for β ensembles on Y. ∗ An,β,Q(z) and R Let N ⊂ Y be a compact neighbourhood of SR . Now −2nQ An,β,Q (z)dτ (z1 ), are, in each variable of the form e |p|β for a W polynomial p ∈ Pn . Hence by repeated use of Theorem 7.2 we have, ( see also section 6) Z

(7.9) and (7.10) Z Z W

−cn

An,β,Q(z)dτ (z) ≤ (1 + O(e Yn

))

Z

An,β,Q(z)dτ (z),

Nn

−cn

An,β,Q (z)dτ (z) ≤ (1 + O(e

))

Y n−1

Z Z W

An,β,Q (z)dτ (z).

N n−1

Note that (7.9) shows that the integrals defining the normalizing constants (1.3) are finite. It also follows that taking N = Yr whether we consider β ensembles on Y or Yr that limn n1 log P robn,β,Q{z1 ∈ W } will be the same. To complete the large deviation property in the unbounded case we must establish exponential tightness. That is : 1 (7.11) lim log P robn,β,Q{|z1 | > r} → −∞ as r → ∞ n n Proceeding as in the proof of Theorem 7.2 but without normalizing the polynomial, we obtain Z Z dτ β −nR(z) β −nR(z) |e p(z)| dτ ≤ ||e p(z)||SR (7.12) nbβ Y \Yr |z| 2 Y \Yr ≤ C||e−nR(z) p(z)||βSR r

−nbβ +a 2

LARGE DEVIATION FOR OUTLYING COORDINATES IN β ENSEMBLES 25

where C is independent of n, p. By the weighted BM inequality, we have Z β −nR(z) n |e−nR p|β dτ (7.13) ||e p(z)||SR ≤ C(1 + ǫ) Yr

≤ C(1 + ǫ)n

Z

|e−nR p|β dτ. Y

Now P robn,β,Q{|z1 | > r} =

1 Znβ,Q

Z

Y \Yr

Z

An,β,Q (z)dτ (z)

Y n−1

and using (7.12) and (7.13) we have n

P robn,β,Q{|z1 | > r} ≤ C(1 + ǫ) r

−nbβ +a 2

1 Znβ,Q

Z

An,β,Q(z)dτ (z). Yn

R 1 A (z)dτ (z) = 1 and since ǫ > 0 is arbitrary, Theorem But Zn,β,Q Y n n,β,Q 7.3 is established.  Remark 7.4. Consider the probability distributions on Y, given by, for W an open subset of Y : ψn′ (W ) := P robn,β,Q{at least one of the coordinates z1 , z2 , . . . , zn ∈ W }. Then the sequence of probability distributions ψn′ under the hypothesis of Theorem 7.3 satisfy the same l.d.p. as ψn . This is because the joint probability distribution is symmetric in z1 , . . . , zn so nP rob{z1 ∈ W } ≥ P rob{at least one of z1 , . . . , zn ∈ W } ≥ P rob{z1 ∈ W }, and the l.d.p for ψn has speed n. References [1] G. Anderson, A. Guionnet and O. Zeitouni, An Introduction to Random Matrices, Cambridge Univ. Press, (2010). [2] T. Bloom, Almost sure convergence for Angelesco ensembles, arXiv:1101.3562v2. [3] T. Bloom, Weighted polynomials and weighted pluripotential theory, Tr. Am. Math. Soc., 361 no.1, (2009), 2163-2179. [4] T. Bloom, Orthogonal polynomials in Cn , Ind. Univ. Math. J., 46 no. 2, (1997), 427-452.

26

THOMAS BLOOM*

[5] T. Bloom and N. Levenberg, Pluripotential energy and large deviation, arXiv:1110.6593v1. To appear in Ind. Univ. Math. J. [6] T. Bloom and N. Levenberg, Asymptotics for Christoffel functions of planar measures, J. d’Anal. Math., 106, (2008), 353-371. [7] T. Bloom and N. Levenberg, Capacity convergence results and applications to a Bernstein-Markov inequality, Tr. Am. Math. Soc., 351 no.12, (1999), 4753-4767. [8] T. Bloom, N. Levenberg and F. Wielonsky, Vector energy and large deviation, arXiv:1301.1095. To appear in J. d’Anal. Math. [9] G. Borot and A. Guionnet, Asymptotic expansion of β matrix models in the one-cut regime, Comm. Math. Phys., 317, (2013), 447-483. [10] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, 2nd edition (1998). [11] P. J. Forrester, Beta ensembles, In: The Oxford Handbook of Random Matrix Theory, Oxford University Press, (2011), 415-432. [12] H. Hedenmalm and N. Makarov, Coulomb gas ensembles and Laplacian growth, Proc. Lond. Math. Soc., (3) 106, (2013), 859-907. [13] F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy, vol. 77, Math. Surveys and Monographs, A.M.S. (2000). [14] K. Johansson, On Fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91, no. 1, (1998), 151-204. [15] M. Klimek, Pluripotential theory, Clarendon Press, (1991). [16] A. B. J. Kuijlaars and K. T-R. McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of a real external fields, Comm on Pure and App. Math, vol LIII, (2000), 736-785. [17] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, (1995). [18] J. Siciak, Extremal plurisubharmonic functions in CN , Ann. Pol. Math., XXXIX, (1981),175-211. [19] E. Saff and V. Totik, Logarithmic potentials with external fields, SpringerVerlag, Berlin, (1997). [20] H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications, vol.43, Cambridge University Press, (1992). University of Toronto, Toronto, Ontario, M5S2E4, Canada E-mail address: [email protected]